Cracking Da Vinci's Code On Tree Growth

Leonardo da Vinci certainly had an eye for detail – and a brain to match. Among his many musings was a theory that a tree’s branches grow in mathematical proportion to the diameter of its trunk. A French researcher now says he has the math to prove Da Vinci was right.

In winter, the sight of trees left bare after its leaves have fallen can leave some feeling gloomy. But for Renaissance genius Leonardo da Vinci, the view was a source of inspiration, which eventually led him to formulate a theory about the relationship between the size of a tree’s truck and the combined measure of its branches. Now, for the first time, a French researcher says he has come up with a scientific explanation for Da Vinci’s 500-year-old theory. In one of his more than 7,000 pages of notes, Da Vinci laid out his tree formula. He calculated that the squared diameter of a tree’s trunk is equal to the squared sum of the diameters of its branches. Moreover, if a trunk is split into two sections, the sum of the thickness of all the branches coming off of one of the section is equal to the thickness of that section of trunk. Up to now, no one had even offered scientific explanation to this theory. Christophe Eloy, a visiting professor at the University of California at San Diego and a specialist in fluid mechanics, may have finally found an answer.

Most botanists thought that the law of nature behind Da Vinci’s formula was an efficient way to transport sap to leaves. Eloy disagrees. He thinks trees are structured as they are in order to protect themselves from damages caused by the wind. Five years ago, Eloy became interested in Leonardo’s formula and the mechanics of trees after reading the book Plaidoyer Pour L'arbre (Plea for the Tree) by Francis Hallé, a specialist in tropical forests. Eloy later attended a lecture by professor of fluid mechanics Emmanuel de Langre at the Ecole Polytechnique in Paris on the relation between wind and plants. “De Langre spoke about Leonardo’s formula, and that it was like a Pythagoras theorem for trees, except that no one had explained it yet,” he recalled.

A question of mechanical engineeringIn his research, Eloy used an analytic model and a numeric model to computer-design the lightest possible tree structure that would still be able to resist the wind. “In the first model, I developed some hypothesis to simplify the geometry of ramification. I took into consideration that a tree’s shape is fractal by nature, meaning that the same pattern repeats itself in smaller structures, such as smaller branches,” says Eloy. “The problem to explain is one of mechanical engineering, to understand how the diameter of the branches should change so that each point of the structure had same probability of breaking when the wind blows.”

The numeric model was used in order to evaluate the effectiveness of some assumptions of the first model. “I used a simple scheme with few parameters to generate the skeleton of the trees in 3-D. Some of the copies of the principal branches were summed up repeatedly in order to create a virtual structure. The diameter of each branch was calculated against the constant of the probability of breaking in the wind,” Eloy says. “The two models perfectly fit with each other and with Leonardo’s formula.” This finding might have several applications, according to the researcher, including a better understanding of the mechanism of how plants grow and resist outside elements. It could be relevant for forestry and agricultural businesses, as well as bioengineering, says Eloy. Such rules of nature occur every day under our eyes. But only Da Vinci, who as both a scientist and a painter, was an attentive enough observer of nature to be able to understand it. “He analyzed and debated his discoveries in a very modern way,” says Eloy. “His intuitions were clear and complex at the same time, and are studied and debated by scientists to this day.”